The function
\begin{equation} f(x) = e^{-a/x} \end{equation}
has a finite value at zero. If I were to Taylor expand $e^{-az}$ and substitute $z = x^{-1}$ I will have a series expansion but it wouldn't be useful since all the terms go as powers of $x^{-1}$. I'm wondering, since this function is finite and analytic near the origin, is there a series representation for this function near $x=0^+$?
For reference, this is what the function looks like for $a=1$:
EDIT:
To clarify, I'm interested in a series representation of the Boltzmann factor $$e^{-H/kT}$$ at low, but finite, temperatures. So I'm interested in the case that $a$ and $x$ are both real and positive.
If we can approximate
$$e^{-1/x}$$
for sufficiently small positive $x$ then you'll have solved your problem.
Define
$$f_n(x)=\begin{cases}\left(1-\frac1{nx}\right)^n,&x>\frac1n\\0,&0\le x\le\frac1n\end{cases}$$
$$g_n(x)=\left(1+\frac1{nx}\right)^{-n}$$
Then,
$$f_n(x)\le e^{-1/x}\le g_n(x),\quad0\le x$$
This approach is effective because
$$1-\frac1{nx}$$
can easily be approximated for small positive $x$ for sufficiently large $n$. An even better approximation would be
$$e^{-1/x}\approx\frac{f_n(x)+g_n(x)}2$$
For example, if you want an error of $E<0.01$, then setting $n=5$ is enough:
$$\left|e^{-1/x}-\frac{f_5(x)+g_5(x)}2\right|<0.01$$